M1.2. Simple Interest Rate

Introduction to Simple Interest

  • Definition:

    • Interest computed as a percentage of the principal.

    • Interest is not reinvested; calculated solely on the initial amount.

  • Key Variables:

    • Principal (PPP): Initial loan or investment amount.

    • Interest rate (rrr): Annual percentage rate, expressed as a decimal or percentage.

    • Time (ttt): Duration of the loan or investment in years.

Simple Interest Formula

I=P×r×t

  • I: Interest earned (in dollars).

  • P: Principal amount.

  • r: Annual interest rate (as a decimal).

  • t: Time in years.

Examples

  1. Annual Interest Example:

    • Sarah deposits 2,000inahigh−interestsavingsaccountpaying42,000 in a high-interest savings account paying 4% annual simple interest.</p></li><li><p>At the end of the first year: I=P×r×t=2000×0.04×1=80I = P \times r \times t = 2000 \times 0.04 \times 1 = 80I=P×r×t=2000×0.04×1=80</p><ul><li><p><strong>Interest earned</strong>:80.

  2. Finding the Principal:

    • How much money must you invest to earn 400ofinterestin200daysat8400 of interest in 200 days at 8% annual interest? I=P×r×t⇒P=Ir×tI = P \times r \times t \quad \Rightarrow \quad P = \frac{I}{r \times t}I=P×r×t⇒P=r×tI​</p><ul><li><p>Convert 200 days to years: t=200365≈0.548t = \frac{200}{365} \approx 0.548t=365200​≈0.548.</p></li><li><p>Substitute values: P=4000.08×0.548P = \frac{400}{0.08 \times 0.548}P=0.08×0.548400​.</p></li></ul></li></ul></li><li><p><strong>Finding the Interest Rate</strong>:</p><ul><li><p>You receive250 interest on a 7,500investmentover4months.Whatistherate?I=P×r×t⇒r=IP×tI=P×r×t⇒r=IP×tI=P×r×t⇒r=P×tI​</p><ul><li><p>Convert4monthstoyears:t=412=13t=412=13t=124​=31​.</p></li><li><p>Substitutevalues:r=2507500×13r=2507500×13r=7500×31​250​.</p></li></ul></li></ul></li><li><p><strong>FindingtheTime</strong>:</p><ul><li><p>Howmanydaysdoesittakefora7,500 investment over 4 months. What is the rate? I=P×r×t⇒r=IP×tI = P \times r \times t \quad \Rightarrow \quad r = \frac{I}{P \times t}I=P×r×t⇒r=P×tI​</p><ul><li><p>Convert 4 months to years: t=412=13t = \frac{4}{12} = \frac{1}{3}t=124​=31​.</p></li><li><p>Substitute values: r=2507500×13r = \frac{250}{7500 \times \frac{1}{3}}r=7500×31​250​.</p></li></ul></li></ul></li><li><p><strong>Finding the Time</strong>:</p><ul><li><p>How many days does it take for a4,300 investment to earn $147 interest at 5.12%? I=P×r×t⇒t=IP×rI = P \times r \times t \quad \Rightarrow \quad t = \frac{I}{P \times r}I=P×r×t⇒t=P×rI​

      • Find ttt in years and convert to days: t×365t \times 365t×365.